关于Borisov-Nuer猜想和Enriques-to-K3映射的象
On the Borisov-Nuer conjecture and the image of the Enriques-to-K3 map
摘要：
讨论了从具有固定极化类型$h$的极化Enriques曲面的模空间$\mathcal M_{En,h}^a$到具有$g=h^2+1$的亏格$g$的极化$K3$曲面的模空间$\mathcal F_g$的正则映射的Borisov-Nuer猜想，并给出了一个自然定义的轨迹$Σ_g\subset\mathcal F_g$ . Borisov-Nuer猜想的一个直接结果是$Σ_g$将包含在$\mathcal F_g$中的一个特定的诺特-莱夫谢茨因子中，我们称之为Borisov-Nuer因子，我们用$\mathcal{BN}_g$表示。本文证明了当$(g-1)$可被$4$整除时，$Σ_g\cap\mathcal{BN}_g$是非空的.为此，我们构造了极化Enriques曲面$(Y, H_Y)$，其中$H_Y^2$可被$4$整除，验证了上述猜想.特别地，如果$h^2$可被$4$整除，并且$h$是相同类型的极化，则该猜想对于任何元素$\mathcal M_{En,h}^a$也成立。
英文摘要：
We discuss the Borisov-Nuer conjecture in connection with the canonical maps from the moduli spaces $\mathcal M_{En,h}^a$of polarized Enriques surfaces with fixed polarization type $h$ to the moduli space $\mathcal F_g$ of polarized $K3$ surfaces of genus $g$ with $g=h^2+1$, and we exhibit a naturally defined locus $Σ_g\subset\mathcal F_g$. One direct consequence of the Borisov-Nuer conjecture is that $Σ_g$ would be contained in a particular Noether-Lefschetz divisor in $\mathcal F_g$, which we call the Borisov-Nuer divisor and we denote by $\mathcal{BN}_g$. In this short note, we prove that $Σ_g\cap\mathcal{BN}_g$ is non-empty whenever $(g-1)$ is divisible by $4$. To this end, we construct polarized Enriques surfaces $(Y, H_Y)$, with $H_Y^2$ divisible by $4$, which verify the conjecture. In particular, the conjecture holds also for any element $\mathcal M_{En,h}^a$, if $h^2$ is divisible by $4$ and $h$ is the same type of polarization.
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